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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.special;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math3.exception.MaxCountExceededException;<a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.exception.NumberIsTooLargeException;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.exception.NumberIsTooSmallException;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.util.ContinuedFraction;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.23"></a>
<FONT color="green">024</FONT>    <a name="line.24"></a>
<FONT color="green">025</FONT>    /**<a name="line.25"></a>
<FONT color="green">026</FONT>     * &lt;p&gt;<a name="line.26"></a>
<FONT color="green">027</FONT>     * This is a utility class that provides computation methods related to the<a name="line.27"></a>
<FONT color="green">028</FONT>     * &amp;Gamma; (Gamma) family of functions.<a name="line.28"></a>
<FONT color="green">029</FONT>     * &lt;/p&gt;<a name="line.29"></a>
<FONT color="green">030</FONT>     * &lt;p&gt;<a name="line.30"></a>
<FONT color="green">031</FONT>     * Implementation of {@link #invGamma1pm1(double)} and<a name="line.31"></a>
<FONT color="green">032</FONT>     * {@link #logGamma1p(double)} is based on the algorithms described in<a name="line.32"></a>
<FONT color="green">033</FONT>     * &lt;ul&gt;<a name="line.33"></a>
<FONT color="green">034</FONT>     * &lt;li&gt;&lt;a href="http://dx.doi.org/10.1145/22721.23109"&gt;Didonato and Morris<a name="line.34"></a>
<FONT color="green">035</FONT>     * (1986)&lt;/a&gt;, &lt;em&gt;Computation of the Incomplete Gamma Function Ratios and<a name="line.35"></a>
<FONT color="green">036</FONT>     *     their Inverse&lt;/em&gt;, TOMS 12(4), 377-393,&lt;/li&gt;<a name="line.36"></a>
<FONT color="green">037</FONT>     * &lt;li&gt;&lt;a href="http://dx.doi.org/10.1145/131766.131776"&gt;Didonato and Morris<a name="line.37"></a>
<FONT color="green">038</FONT>     * (1992)&lt;/a&gt;, &lt;em&gt;Algorithm 708: Significant Digit Computation of the<a name="line.38"></a>
<FONT color="green">039</FONT>     *     Incomplete Beta Function Ratios&lt;/em&gt;, TOMS 18(3), 360-373,&lt;/li&gt;<a name="line.39"></a>
<FONT color="green">040</FONT>     * &lt;/ul&gt;<a name="line.40"></a>
<FONT color="green">041</FONT>     * and implemented in the<a name="line.41"></a>
<FONT color="green">042</FONT>     * &lt;a href="http://www.dtic.mil/docs/citations/ADA476840"&gt;NSWC Library of Mathematical Functions&lt;/a&gt;,<a name="line.42"></a>
<FONT color="green">043</FONT>     * available<a name="line.43"></a>
<FONT color="green">044</FONT>     * &lt;a href="http://www.ualberta.ca/CNS/RESEARCH/Software/NumericalNSWC/site.html"&gt;here&lt;/a&gt;.<a name="line.44"></a>
<FONT color="green">045</FONT>     * This library is "approved for public release", and the<a name="line.45"></a>
<FONT color="green">046</FONT>     * &lt;a href="http://www.dtic.mil/dtic/pdf/announcements/CopyrightGuidance.pdf"&gt;Copyright guidance&lt;/a&gt;<a name="line.46"></a>
<FONT color="green">047</FONT>     * indicates that unless otherwise stated in the code, all FORTRAN functions in<a name="line.47"></a>
<FONT color="green">048</FONT>     * this library are license free. Since no such notice appears in the code these<a name="line.48"></a>
<FONT color="green">049</FONT>     * functions can safely be ported to Commons-Math.<a name="line.49"></a>
<FONT color="green">050</FONT>     * &lt;/p&gt;<a name="line.50"></a>
<FONT color="green">051</FONT>     *<a name="line.51"></a>
<FONT color="green">052</FONT>     * @version $Id: Gamma.java 1422313 2012-12-15 18:53:41Z psteitz $<a name="line.52"></a>
<FONT color="green">053</FONT>     */<a name="line.53"></a>
<FONT color="green">054</FONT>    public class Gamma {<a name="line.54"></a>
<FONT color="green">055</FONT>        /**<a name="line.55"></a>
<FONT color="green">056</FONT>         * &lt;a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant"&gt;Euler-Mascheroni constant&lt;/a&gt;<a name="line.56"></a>
<FONT color="green">057</FONT>         * @since 2.0<a name="line.57"></a>
<FONT color="green">058</FONT>         */<a name="line.58"></a>
<FONT color="green">059</FONT>        public static final double GAMMA = 0.577215664901532860606512090082;<a name="line.59"></a>
<FONT color="green">060</FONT>    <a name="line.60"></a>
<FONT color="green">061</FONT>        /**<a name="line.61"></a>
<FONT color="green">062</FONT>         * The value of the {@code g} constant in the Lanczos approximation, see<a name="line.62"></a>
<FONT color="green">063</FONT>         * {@link #lanczos(double)}.<a name="line.63"></a>
<FONT color="green">064</FONT>         * @since 3.1<a name="line.64"></a>
<FONT color="green">065</FONT>         */<a name="line.65"></a>
<FONT color="green">066</FONT>        public static final double LANCZOS_G = 607.0 / 128.0;<a name="line.66"></a>
<FONT color="green">067</FONT>    <a name="line.67"></a>
<FONT color="green">068</FONT>        /** Maximum allowed numerical error. */<a name="line.68"></a>
<FONT color="green">069</FONT>        private static final double DEFAULT_EPSILON = 10e-15;<a name="line.69"></a>
<FONT color="green">070</FONT>    <a name="line.70"></a>
<FONT color="green">071</FONT>        /** Lanczos coefficients */<a name="line.71"></a>
<FONT color="green">072</FONT>        private static final double[] LANCZOS = {<a name="line.72"></a>
<FONT color="green">073</FONT>            0.99999999999999709182,<a name="line.73"></a>
<FONT color="green">074</FONT>            57.156235665862923517,<a name="line.74"></a>
<FONT color="green">075</FONT>            -59.597960355475491248,<a name="line.75"></a>
<FONT color="green">076</FONT>            14.136097974741747174,<a name="line.76"></a>
<FONT color="green">077</FONT>            -0.49191381609762019978,<a name="line.77"></a>
<FONT color="green">078</FONT>            .33994649984811888699e-4,<a name="line.78"></a>
<FONT color="green">079</FONT>            .46523628927048575665e-4,<a name="line.79"></a>
<FONT color="green">080</FONT>            -.98374475304879564677e-4,<a name="line.80"></a>
<FONT color="green">081</FONT>            .15808870322491248884e-3,<a name="line.81"></a>
<FONT color="green">082</FONT>            -.21026444172410488319e-3,<a name="line.82"></a>
<FONT color="green">083</FONT>            .21743961811521264320e-3,<a name="line.83"></a>
<FONT color="green">084</FONT>            -.16431810653676389022e-3,<a name="line.84"></a>
<FONT color="green">085</FONT>            .84418223983852743293e-4,<a name="line.85"></a>
<FONT color="green">086</FONT>            -.26190838401581408670e-4,<a name="line.86"></a>
<FONT color="green">087</FONT>            .36899182659531622704e-5,<a name="line.87"></a>
<FONT color="green">088</FONT>        };<a name="line.88"></a>
<FONT color="green">089</FONT>    <a name="line.89"></a>
<FONT color="green">090</FONT>        /** Avoid repeated computation of log of 2 PI in logGamma */<a name="line.90"></a>
<FONT color="green">091</FONT>        private static final double HALF_LOG_2_PI = 0.5 * FastMath.log(2.0 * FastMath.PI);<a name="line.91"></a>
<FONT color="green">092</FONT>    <a name="line.92"></a>
<FONT color="green">093</FONT>        /** The constant value of &amp;radic;(2&amp;pi;). */<a name="line.93"></a>
<FONT color="green">094</FONT>        private static final double SQRT_TWO_PI = 2.506628274631000502;<a name="line.94"></a>
<FONT color="green">095</FONT>    <a name="line.95"></a>
<FONT color="green">096</FONT>        // limits for switching algorithm in digamma<a name="line.96"></a>
<FONT color="green">097</FONT>        /** C limit. */<a name="line.97"></a>
<FONT color="green">098</FONT>        private static final double C_LIMIT = 49;<a name="line.98"></a>
<FONT color="green">099</FONT>    <a name="line.99"></a>
<FONT color="green">100</FONT>        /** S limit. */<a name="line.100"></a>
<FONT color="green">101</FONT>        private static final double S_LIMIT = 1e-5;<a name="line.101"></a>
<FONT color="green">102</FONT>    <a name="line.102"></a>
<FONT color="green">103</FONT>        /*<a name="line.103"></a>
<FONT color="green">104</FONT>         * Constants for the computation of double invGamma1pm1(double).<a name="line.104"></a>
<FONT color="green">105</FONT>         * Copied from DGAM1 in the NSWC library.<a name="line.105"></a>
<FONT color="green">106</FONT>         */<a name="line.106"></a>
<FONT color="green">107</FONT>    <a name="line.107"></a>
<FONT color="green">108</FONT>        /** The constant {@code A0} defined in {@code DGAM1}. */<a name="line.108"></a>
<FONT color="green">109</FONT>        private static final double INV_GAMMA1P_M1_A0 = .611609510448141581788E-08;<a name="line.109"></a>
<FONT color="green">110</FONT>    <a name="line.110"></a>
<FONT color="green">111</FONT>        /** The constant {@code A1} defined in {@code DGAM1}. */<a name="line.111"></a>
<FONT color="green">112</FONT>        private static final double INV_GAMMA1P_M1_A1 = .624730830116465516210E-08;<a name="line.112"></a>
<FONT color="green">113</FONT>    <a name="line.113"></a>
<FONT color="green">114</FONT>        /** The constant {@code B1} defined in {@code DGAM1}. */<a name="line.114"></a>
<FONT color="green">115</FONT>        private static final double INV_GAMMA1P_M1_B1 = .203610414066806987300E+00;<a name="line.115"></a>
<FONT color="green">116</FONT>    <a name="line.116"></a>
<FONT color="green">117</FONT>        /** The constant {@code B2} defined in {@code DGAM1}. */<a name="line.117"></a>
<FONT color="green">118</FONT>        private static final double INV_GAMMA1P_M1_B2 = .266205348428949217746E-01;<a name="line.118"></a>
<FONT color="green">119</FONT>    <a name="line.119"></a>
<FONT color="green">120</FONT>        /** The constant {@code B3} defined in {@code DGAM1}. */<a name="line.120"></a>
<FONT color="green">121</FONT>        private static final double INV_GAMMA1P_M1_B3 = .493944979382446875238E-03;<a name="line.121"></a>
<FONT color="green">122</FONT>    <a name="line.122"></a>
<FONT color="green">123</FONT>        /** The constant {@code B4} defined in {@code DGAM1}. */<a name="line.123"></a>
<FONT color="green">124</FONT>        private static final double INV_GAMMA1P_M1_B4 = -.851419432440314906588E-05;<a name="line.124"></a>
<FONT color="green">125</FONT>    <a name="line.125"></a>
<FONT color="green">126</FONT>        /** The constant {@code B5} defined in {@code DGAM1}. */<a name="line.126"></a>
<FONT color="green">127</FONT>        private static final double INV_GAMMA1P_M1_B5 = -.643045481779353022248E-05;<a name="line.127"></a>
<FONT color="green">128</FONT>    <a name="line.128"></a>
<FONT color="green">129</FONT>        /** The constant {@code B6} defined in {@code DGAM1}. */<a name="line.129"></a>
<FONT color="green">130</FONT>        private static final double INV_GAMMA1P_M1_B6 = .992641840672773722196E-06;<a name="line.130"></a>
<FONT color="green">131</FONT>    <a name="line.131"></a>
<FONT color="green">132</FONT>        /** The constant {@code B7} defined in {@code DGAM1}. */<a name="line.132"></a>
<FONT color="green">133</FONT>        private static final double INV_GAMMA1P_M1_B7 = -.607761895722825260739E-07;<a name="line.133"></a>
<FONT color="green">134</FONT>    <a name="line.134"></a>
<FONT color="green">135</FONT>        /** The constant {@code B8} defined in {@code DGAM1}. */<a name="line.135"></a>
<FONT color="green">136</FONT>        private static final double INV_GAMMA1P_M1_B8 = .195755836614639731882E-09;<a name="line.136"></a>
<FONT color="green">137</FONT>    <a name="line.137"></a>
<FONT color="green">138</FONT>        /** The constant {@code P0} defined in {@code DGAM1}. */<a name="line.138"></a>
<FONT color="green">139</FONT>        private static final double INV_GAMMA1P_M1_P0 = .6116095104481415817861E-08;<a name="line.139"></a>
<FONT color="green">140</FONT>    <a name="line.140"></a>
<FONT color="green">141</FONT>        /** The constant {@code P1} defined in {@code DGAM1}. */<a name="line.141"></a>
<FONT color="green">142</FONT>        private static final double INV_GAMMA1P_M1_P1 = .6871674113067198736152E-08;<a name="line.142"></a>
<FONT color="green">143</FONT>    <a name="line.143"></a>
<FONT color="green">144</FONT>        /** The constant {@code P2} defined in {@code DGAM1}. */<a name="line.144"></a>
<FONT color="green">145</FONT>        private static final double INV_GAMMA1P_M1_P2 = .6820161668496170657918E-09;<a name="line.145"></a>
<FONT color="green">146</FONT>    <a name="line.146"></a>
<FONT color="green">147</FONT>        /** The constant {@code P3} defined in {@code DGAM1}. */<a name="line.147"></a>
<FONT color="green">148</FONT>        private static final double INV_GAMMA1P_M1_P3 = .4686843322948848031080E-10;<a name="line.148"></a>
<FONT color="green">149</FONT>    <a name="line.149"></a>
<FONT color="green">150</FONT>        /** The constant {@code P4} defined in {@code DGAM1}. */<a name="line.150"></a>
<FONT color="green">151</FONT>        private static final double INV_GAMMA1P_M1_P4 = .1572833027710446286995E-11;<a name="line.151"></a>
<FONT color="green">152</FONT>    <a name="line.152"></a>
<FONT color="green">153</FONT>        /** The constant {@code P5} defined in {@code DGAM1}. */<a name="line.153"></a>
<FONT color="green">154</FONT>        private static final double INV_GAMMA1P_M1_P5 = -.1249441572276366213222E-12;<a name="line.154"></a>
<FONT color="green">155</FONT>    <a name="line.155"></a>
<FONT color="green">156</FONT>        /** The constant {@code P6} defined in {@code DGAM1}. */<a name="line.156"></a>
<FONT color="green">157</FONT>        private static final double INV_GAMMA1P_M1_P6 = .4343529937408594255178E-14;<a name="line.157"></a>
<FONT color="green">158</FONT>    <a name="line.158"></a>
<FONT color="green">159</FONT>        /** The constant {@code Q1} defined in {@code DGAM1}. */<a name="line.159"></a>
<FONT color="green">160</FONT>        private static final double INV_GAMMA1P_M1_Q1 = .3056961078365221025009E+00;<a name="line.160"></a>
<FONT color="green">161</FONT>    <a name="line.161"></a>
<FONT color="green">162</FONT>        /** The constant {@code Q2} defined in {@code DGAM1}. */<a name="line.162"></a>
<FONT color="green">163</FONT>        private static final double INV_GAMMA1P_M1_Q2 = .5464213086042296536016E-01;<a name="line.163"></a>
<FONT color="green">164</FONT>    <a name="line.164"></a>
<FONT color="green">165</FONT>        /** The constant {@code Q3} defined in {@code DGAM1}. */<a name="line.165"></a>
<FONT color="green">166</FONT>        private static final double INV_GAMMA1P_M1_Q3 = .4956830093825887312020E-02;<a name="line.166"></a>
<FONT color="green">167</FONT>    <a name="line.167"></a>
<FONT color="green">168</FONT>        /** The constant {@code Q4} defined in {@code DGAM1}. */<a name="line.168"></a>
<FONT color="green">169</FONT>        private static final double INV_GAMMA1P_M1_Q4 = .2692369466186361192876E-03;<a name="line.169"></a>
<FONT color="green">170</FONT>    <a name="line.170"></a>
<FONT color="green">171</FONT>        /** The constant {@code C} defined in {@code DGAM1}. */<a name="line.171"></a>
<FONT color="green">172</FONT>        private static final double INV_GAMMA1P_M1_C = -.422784335098467139393487909917598E+00;<a name="line.172"></a>
<FONT color="green">173</FONT>    <a name="line.173"></a>
<FONT color="green">174</FONT>        /** The constant {@code C0} defined in {@code DGAM1}. */<a name="line.174"></a>
<FONT color="green">175</FONT>        private static final double INV_GAMMA1P_M1_C0 = .577215664901532860606512090082402E+00;<a name="line.175"></a>
<FONT color="green">176</FONT>    <a name="line.176"></a>
<FONT color="green">177</FONT>        /** The constant {@code C1} defined in {@code DGAM1}. */<a name="line.177"></a>
<FONT color="green">178</FONT>        private static final double INV_GAMMA1P_M1_C1 = -.655878071520253881077019515145390E+00;<a name="line.178"></a>
<FONT color="green">179</FONT>    <a name="line.179"></a>
<FONT color="green">180</FONT>        /** The constant {@code C2} defined in {@code DGAM1}. */<a name="line.180"></a>
<FONT color="green">181</FONT>        private static final double INV_GAMMA1P_M1_C2 = -.420026350340952355290039348754298E-01;<a name="line.181"></a>
<FONT color="green">182</FONT>    <a name="line.182"></a>
<FONT color="green">183</FONT>        /** The constant {@code C3} defined in {@code DGAM1}. */<a name="line.183"></a>
<FONT color="green">184</FONT>        private static final double INV_GAMMA1P_M1_C3 = .166538611382291489501700795102105E+00;<a name="line.184"></a>
<FONT color="green">185</FONT>    <a name="line.185"></a>
<FONT color="green">186</FONT>        /** The constant {@code C4} defined in {@code DGAM1}. */<a name="line.186"></a>
<FONT color="green">187</FONT>        private static final double INV_GAMMA1P_M1_C4 = -.421977345555443367482083012891874E-01;<a name="line.187"></a>
<FONT color="green">188</FONT>    <a name="line.188"></a>
<FONT color="green">189</FONT>        /** The constant {@code C5} defined in {@code DGAM1}. */<a name="line.189"></a>
<FONT color="green">190</FONT>        private static final double INV_GAMMA1P_M1_C5 = -.962197152787697356211492167234820E-02;<a name="line.190"></a>
<FONT color="green">191</FONT>    <a name="line.191"></a>
<FONT color="green">192</FONT>        /** The constant {@code C6} defined in {@code DGAM1}. */<a name="line.192"></a>
<FONT color="green">193</FONT>        private static final double INV_GAMMA1P_M1_C6 = .721894324666309954239501034044657E-02;<a name="line.193"></a>
<FONT color="green">194</FONT>    <a name="line.194"></a>
<FONT color="green">195</FONT>        /** The constant {@code C7} defined in {@code DGAM1}. */<a name="line.195"></a>
<FONT color="green">196</FONT>        private static final double INV_GAMMA1P_M1_C7 = -.116516759185906511211397108401839E-02;<a name="line.196"></a>
<FONT color="green">197</FONT>    <a name="line.197"></a>
<FONT color="green">198</FONT>        /** The constant {@code C8} defined in {@code DGAM1}. */<a name="line.198"></a>
<FONT color="green">199</FONT>        private static final double INV_GAMMA1P_M1_C8 = -.215241674114950972815729963053648E-03;<a name="line.199"></a>
<FONT color="green">200</FONT>    <a name="line.200"></a>
<FONT color="green">201</FONT>        /** The constant {@code C9} defined in {@code DGAM1}. */<a name="line.201"></a>
<FONT color="green">202</FONT>        private static final double INV_GAMMA1P_M1_C9 = .128050282388116186153198626328164E-03;<a name="line.202"></a>
<FONT color="green">203</FONT>    <a name="line.203"></a>
<FONT color="green">204</FONT>        /** The constant {@code C10} defined in {@code DGAM1}. */<a name="line.204"></a>
<FONT color="green">205</FONT>        private static final double INV_GAMMA1P_M1_C10 = -.201348547807882386556893914210218E-04;<a name="line.205"></a>
<FONT color="green">206</FONT>    <a name="line.206"></a>
<FONT color="green">207</FONT>        /** The constant {@code C11} defined in {@code DGAM1}. */<a name="line.207"></a>
<FONT color="green">208</FONT>        private static final double INV_GAMMA1P_M1_C11 = -.125049348214267065734535947383309E-05;<a name="line.208"></a>
<FONT color="green">209</FONT>    <a name="line.209"></a>
<FONT color="green">210</FONT>        /** The constant {@code C12} defined in {@code DGAM1}. */<a name="line.210"></a>
<FONT color="green">211</FONT>        private static final double INV_GAMMA1P_M1_C12 = .113302723198169588237412962033074E-05;<a name="line.211"></a>
<FONT color="green">212</FONT>    <a name="line.212"></a>
<FONT color="green">213</FONT>        /** The constant {@code C13} defined in {@code DGAM1}. */<a name="line.213"></a>
<FONT color="green">214</FONT>        private static final double INV_GAMMA1P_M1_C13 = -.205633841697760710345015413002057E-06;<a name="line.214"></a>
<FONT color="green">215</FONT>    <a name="line.215"></a>
<FONT color="green">216</FONT>        /**<a name="line.216"></a>
<FONT color="green">217</FONT>         * Default constructor.  Prohibit instantiation.<a name="line.217"></a>
<FONT color="green">218</FONT>         */<a name="line.218"></a>
<FONT color="green">219</FONT>        private Gamma() {}<a name="line.219"></a>
<FONT color="green">220</FONT>    <a name="line.220"></a>
<FONT color="green">221</FONT>        /**<a name="line.221"></a>
<FONT color="green">222</FONT>         * &lt;p&gt;<a name="line.222"></a>
<FONT color="green">223</FONT>         * Returns the value of log&amp;nbsp;&amp;Gamma;(x) for x&amp;nbsp;&amp;gt;&amp;nbsp;0.<a name="line.223"></a>
<FONT color="green">224</FONT>         * &lt;/p&gt;<a name="line.224"></a>
<FONT color="green">225</FONT>         * &lt;p&gt;<a name="line.225"></a>
<FONT color="green">226</FONT>         * For x &amp;le; 8, the implementation is based on the double precision<a name="line.226"></a>
<FONT color="green">227</FONT>         * implementation in the &lt;em&gt;NSWC Library of Mathematics Subroutines&lt;/em&gt;,<a name="line.227"></a>
<FONT color="green">228</FONT>         * {@code DGAMLN}. For x &amp;gt; 8, the implementation is based on<a name="line.228"></a>
<FONT color="green">229</FONT>         * &lt;/p&gt;<a name="line.229"></a>
<FONT color="green">230</FONT>         * &lt;ul&gt;<a name="line.230"></a>
<FONT color="green">231</FONT>         * &lt;li&gt;&lt;a href="http://mathworld.wolfram.com/GammaFunction.html"&gt;Gamma<a name="line.231"></a>
<FONT color="green">232</FONT>         *     Function&lt;/a&gt;, equation (28).&lt;/li&gt;<a name="line.232"></a>
<FONT color="green">233</FONT>         * &lt;li&gt;&lt;a href="http://mathworld.wolfram.com/LanczosApproximation.html"&gt;<a name="line.233"></a>
<FONT color="green">234</FONT>         *     Lanczos Approximation&lt;/a&gt;, equations (1) through (5).&lt;/li&gt;<a name="line.234"></a>
<FONT color="green">235</FONT>         * &lt;li&gt;&lt;a href="http://my.fit.edu/~gabdo/gamma.txt"&gt;Paul Godfrey, A note on<a name="line.235"></a>
<FONT color="green">236</FONT>         *     the computation of the convergent Lanczos complex Gamma<a name="line.236"></a>
<FONT color="green">237</FONT>         *     approximation&lt;/a&gt;&lt;/li&gt;<a name="line.237"></a>
<FONT color="green">238</FONT>         * &lt;/ul&gt;<a name="line.238"></a>
<FONT color="green">239</FONT>         *<a name="line.239"></a>
<FONT color="green">240</FONT>         * @param x Argument.<a name="line.240"></a>
<FONT color="green">241</FONT>         * @return the value of {@code log(Gamma(x))}, {@code Double.NaN} if<a name="line.241"></a>
<FONT color="green">242</FONT>         * {@code x &lt;= 0.0}.<a name="line.242"></a>
<FONT color="green">243</FONT>         */<a name="line.243"></a>
<FONT color="green">244</FONT>        public static double logGamma(double x) {<a name="line.244"></a>
<FONT color="green">245</FONT>            double ret;<a name="line.245"></a>
<FONT color="green">246</FONT>    <a name="line.246"></a>
<FONT color="green">247</FONT>            if (Double.isNaN(x) || (x &lt;= 0.0)) {<a name="line.247"></a>
<FONT color="green">248</FONT>                ret = Double.NaN;<a name="line.248"></a>
<FONT color="green">249</FONT>            } else if (x &lt; 0.5) {<a name="line.249"></a>
<FONT color="green">250</FONT>                return logGamma1p(x) - FastMath.log(x);<a name="line.250"></a>
<FONT color="green">251</FONT>            } else if (x &lt;= 2.5) {<a name="line.251"></a>
<FONT color="green">252</FONT>                return logGamma1p((x - 0.5) - 0.5);<a name="line.252"></a>
<FONT color="green">253</FONT>            } else if (x &lt;= 8.0) {<a name="line.253"></a>
<FONT color="green">254</FONT>                final int n = (int) FastMath.floor(x - 1.5);<a name="line.254"></a>
<FONT color="green">255</FONT>                double prod = 1.0;<a name="line.255"></a>
<FONT color="green">256</FONT>                for (int i = 1; i &lt;= n; i++) {<a name="line.256"></a>
<FONT color="green">257</FONT>                    prod *= x - i;<a name="line.257"></a>
<FONT color="green">258</FONT>                }<a name="line.258"></a>
<FONT color="green">259</FONT>                return logGamma1p(x - (n + 1)) + FastMath.log(prod);<a name="line.259"></a>
<FONT color="green">260</FONT>            } else {<a name="line.260"></a>
<FONT color="green">261</FONT>                double sum = lanczos(x);<a name="line.261"></a>
<FONT color="green">262</FONT>                double tmp = x + LANCZOS_G + .5;<a name="line.262"></a>
<FONT color="green">263</FONT>                ret = ((x + .5) * FastMath.log(tmp)) - tmp +<a name="line.263"></a>
<FONT color="green">264</FONT>                    HALF_LOG_2_PI + FastMath.log(sum / x);<a name="line.264"></a>
<FONT color="green">265</FONT>            }<a name="line.265"></a>
<FONT color="green">266</FONT>    <a name="line.266"></a>
<FONT color="green">267</FONT>            return ret;<a name="line.267"></a>
<FONT color="green">268</FONT>        }<a name="line.268"></a>
<FONT color="green">269</FONT>    <a name="line.269"></a>
<FONT color="green">270</FONT>        /**<a name="line.270"></a>
<FONT color="green">271</FONT>         * Returns the regularized gamma function P(a, x).<a name="line.271"></a>
<FONT color="green">272</FONT>         *<a name="line.272"></a>
<FONT color="green">273</FONT>         * @param a Parameter.<a name="line.273"></a>
<FONT color="green">274</FONT>         * @param x Value.<a name="line.274"></a>
<FONT color="green">275</FONT>         * @return the regularized gamma function P(a, x).<a name="line.275"></a>
<FONT color="green">276</FONT>         * @throws MaxCountExceededException if the algorithm fails to converge.<a name="line.276"></a>
<FONT color="green">277</FONT>         */<a name="line.277"></a>
<FONT color="green">278</FONT>        public static double regularizedGammaP(double a, double x) {<a name="line.278"></a>
<FONT color="green">279</FONT>            return regularizedGammaP(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);<a name="line.279"></a>
<FONT color="green">280</FONT>        }<a name="line.280"></a>
<FONT color="green">281</FONT>    <a name="line.281"></a>
<FONT color="green">282</FONT>        /**<a name="line.282"></a>
<FONT color="green">283</FONT>         * Returns the regularized gamma function P(a, x).<a name="line.283"></a>
<FONT color="green">284</FONT>         *<a name="line.284"></a>
<FONT color="green">285</FONT>         * The implementation of this method is based on:<a name="line.285"></a>
<FONT color="green">286</FONT>         * &lt;ul&gt;<a name="line.286"></a>
<FONT color="green">287</FONT>         *  &lt;li&gt;<a name="line.287"></a>
<FONT color="green">288</FONT>         *   &lt;a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html"&gt;<a name="line.288"></a>
<FONT color="green">289</FONT>         *   Regularized Gamma Function&lt;/a&gt;, equation (1)<a name="line.289"></a>
<FONT color="green">290</FONT>         *  &lt;/li&gt;<a name="line.290"></a>
<FONT color="green">291</FONT>         *  &lt;li&gt;<a name="line.291"></a>
<FONT color="green">292</FONT>         *   &lt;a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html"&gt;<a name="line.292"></a>
<FONT color="green">293</FONT>         *   Incomplete Gamma Function&lt;/a&gt;, equation (4).<a name="line.293"></a>
<FONT color="green">294</FONT>         *  &lt;/li&gt;<a name="line.294"></a>
<FONT color="green">295</FONT>         *  &lt;li&gt;<a name="line.295"></a>
<FONT color="green">296</FONT>         *   &lt;a href="http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html"&gt;<a name="line.296"></a>
<FONT color="green">297</FONT>         *   Confluent Hypergeometric Function of the First Kind&lt;/a&gt;, equation (1).<a name="line.297"></a>
<FONT color="green">298</FONT>         *  &lt;/li&gt;<a name="line.298"></a>
<FONT color="green">299</FONT>         * &lt;/ul&gt;<a name="line.299"></a>
<FONT color="green">300</FONT>         *<a name="line.300"></a>
<FONT color="green">301</FONT>         * @param a the a parameter.<a name="line.301"></a>
<FONT color="green">302</FONT>         * @param x the value.<a name="line.302"></a>
<FONT color="green">303</FONT>         * @param epsilon When the absolute value of the nth item in the<a name="line.303"></a>
<FONT color="green">304</FONT>         * series is less than epsilon the approximation ceases to calculate<a name="line.304"></a>
<FONT color="green">305</FONT>         * further elements in the series.<a name="line.305"></a>
<FONT color="green">306</FONT>         * @param maxIterations Maximum number of "iterations" to complete.<a name="line.306"></a>
<FONT color="green">307</FONT>         * @return the regularized gamma function P(a, x)<a name="line.307"></a>
<FONT color="green">308</FONT>         * @throws MaxCountExceededException if the algorithm fails to converge.<a name="line.308"></a>
<FONT color="green">309</FONT>         */<a name="line.309"></a>
<FONT color="green">310</FONT>        public static double regularizedGammaP(double a,<a name="line.310"></a>
<FONT color="green">311</FONT>                                               double x,<a name="line.311"></a>
<FONT color="green">312</FONT>                                               double epsilon,<a name="line.312"></a>
<FONT color="green">313</FONT>                                               int maxIterations) {<a name="line.313"></a>
<FONT color="green">314</FONT>            double ret;<a name="line.314"></a>
<FONT color="green">315</FONT>    <a name="line.315"></a>
<FONT color="green">316</FONT>            if (Double.isNaN(a) || Double.isNaN(x) || (a &lt;= 0.0) || (x &lt; 0.0)) {<a name="line.316"></a>
<FONT color="green">317</FONT>                ret = Double.NaN;<a name="line.317"></a>
<FONT color="green">318</FONT>            } else if (x == 0.0) {<a name="line.318"></a>
<FONT color="green">319</FONT>                ret = 0.0;<a name="line.319"></a>
<FONT color="green">320</FONT>            } else if (x &gt;= a + 1) {<a name="line.320"></a>
<FONT color="green">321</FONT>                // use regularizedGammaQ because it should converge faster in this<a name="line.321"></a>
<FONT color="green">322</FONT>                // case.<a name="line.322"></a>
<FONT color="green">323</FONT>                ret = 1.0 - regularizedGammaQ(a, x, epsilon, maxIterations);<a name="line.323"></a>
<FONT color="green">324</FONT>            } else {<a name="line.324"></a>
<FONT color="green">325</FONT>                // calculate series<a name="line.325"></a>
<FONT color="green">326</FONT>                double n = 0.0; // current element index<a name="line.326"></a>
<FONT color="green">327</FONT>                double an = 1.0 / a; // n-th element in the series<a name="line.327"></a>
<FONT color="green">328</FONT>                double sum = an; // partial sum<a name="line.328"></a>
<FONT color="green">329</FONT>                while (FastMath.abs(an/sum) &gt; epsilon &amp;&amp;<a name="line.329"></a>
<FONT color="green">330</FONT>                       n &lt; maxIterations &amp;&amp;<a name="line.330"></a>
<FONT color="green">331</FONT>                       sum &lt; Double.POSITIVE_INFINITY) {<a name="line.331"></a>
<FONT color="green">332</FONT>                    // compute next element in the series<a name="line.332"></a>
<FONT color="green">333</FONT>                    n = n + 1.0;<a name="line.333"></a>
<FONT color="green">334</FONT>                    an = an * (x / (a + n));<a name="line.334"></a>
<FONT color="green">335</FONT>    <a name="line.335"></a>
<FONT color="green">336</FONT>                    // update partial sum<a name="line.336"></a>
<FONT color="green">337</FONT>                    sum = sum + an;<a name="line.337"></a>
<FONT color="green">338</FONT>                }<a name="line.338"></a>
<FONT color="green">339</FONT>                if (n &gt;= maxIterations) {<a name="line.339"></a>
<FONT color="green">340</FONT>                    throw new MaxCountExceededException(maxIterations);<a name="line.340"></a>
<FONT color="green">341</FONT>                } else if (Double.isInfinite(sum)) {<a name="line.341"></a>
<FONT color="green">342</FONT>                    ret = 1.0;<a name="line.342"></a>
<FONT color="green">343</FONT>                } else {<a name="line.343"></a>
<FONT color="green">344</FONT>                    ret = FastMath.exp(-x + (a * FastMath.log(x)) - logGamma(a)) * sum;<a name="line.344"></a>
<FONT color="green">345</FONT>                }<a name="line.345"></a>
<FONT color="green">346</FONT>            }<a name="line.346"></a>
<FONT color="green">347</FONT>    <a name="line.347"></a>
<FONT color="green">348</FONT>            return ret;<a name="line.348"></a>
<FONT color="green">349</FONT>        }<a name="line.349"></a>
<FONT color="green">350</FONT>    <a name="line.350"></a>
<FONT color="green">351</FONT>        /**<a name="line.351"></a>
<FONT color="green">352</FONT>         * Returns the regularized gamma function Q(a, x) = 1 - P(a, x).<a name="line.352"></a>
<FONT color="green">353</FONT>         *<a name="line.353"></a>
<FONT color="green">354</FONT>         * @param a the a parameter.<a name="line.354"></a>
<FONT color="green">355</FONT>         * @param x the value.<a name="line.355"></a>
<FONT color="green">356</FONT>         * @return the regularized gamma function Q(a, x)<a name="line.356"></a>
<FONT color="green">357</FONT>         * @throws MaxCountExceededException if the algorithm fails to converge.<a name="line.357"></a>
<FONT color="green">358</FONT>         */<a name="line.358"></a>
<FONT color="green">359</FONT>        public static double regularizedGammaQ(double a, double x) {<a name="line.359"></a>
<FONT color="green">360</FONT>            return regularizedGammaQ(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);<a name="line.360"></a>
<FONT color="green">361</FONT>        }<a name="line.361"></a>
<FONT color="green">362</FONT>    <a name="line.362"></a>
<FONT color="green">363</FONT>        /**<a name="line.363"></a>
<FONT color="green">364</FONT>         * Returns the regularized gamma function Q(a, x) = 1 - P(a, x).<a name="line.364"></a>
<FONT color="green">365</FONT>         *<a name="line.365"></a>
<FONT color="green">366</FONT>         * The implementation of this method is based on:<a name="line.366"></a>
<FONT color="green">367</FONT>         * &lt;ul&gt;<a name="line.367"></a>
<FONT color="green">368</FONT>         *  &lt;li&gt;<a name="line.368"></a>
<FONT color="green">369</FONT>         *   &lt;a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html"&gt;<a name="line.369"></a>
<FONT color="green">370</FONT>         *   Regularized Gamma Function&lt;/a&gt;, equation (1).<a name="line.370"></a>
<FONT color="green">371</FONT>         *  &lt;/li&gt;<a name="line.371"></a>
<FONT color="green">372</FONT>         *  &lt;li&gt;<a name="line.372"></a>
<FONT color="green">373</FONT>         *   &lt;a href="http://functions.wolfram.com/GammaBetaErf/GammaRegularized/10/0003/"&gt;<a name="line.373"></a>
<FONT color="green">374</FONT>         *   Regularized incomplete gamma function: Continued fraction representations<a name="line.374"></a>
<FONT color="green">375</FONT>         *   (formula 06.08.10.0003)&lt;/a&gt;<a name="line.375"></a>
<FONT color="green">376</FONT>         *  &lt;/li&gt;<a name="line.376"></a>
<FONT color="green">377</FONT>         * &lt;/ul&gt;<a name="line.377"></a>
<FONT color="green">378</FONT>         *<a name="line.378"></a>
<FONT color="green">379</FONT>         * @param a the a parameter.<a name="line.379"></a>
<FONT color="green">380</FONT>         * @param x the value.<a name="line.380"></a>
<FONT color="green">381</FONT>         * @param epsilon When the absolute value of the nth item in the<a name="line.381"></a>
<FONT color="green">382</FONT>         * series is less than epsilon the approximation ceases to calculate<a name="line.382"></a>
<FONT color="green">383</FONT>         * further elements in the series.<a name="line.383"></a>
<FONT color="green">384</FONT>         * @param maxIterations Maximum number of "iterations" to complete.<a name="line.384"></a>
<FONT color="green">385</FONT>         * @return the regularized gamma function P(a, x)<a name="line.385"></a>
<FONT color="green">386</FONT>         * @throws MaxCountExceededException if the algorithm fails to converge.<a name="line.386"></a>
<FONT color="green">387</FONT>         */<a name="line.387"></a>
<FONT color="green">388</FONT>        public static double regularizedGammaQ(final double a,<a name="line.388"></a>
<FONT color="green">389</FONT>                                               double x,<a name="line.389"></a>
<FONT color="green">390</FONT>                                               double epsilon,<a name="line.390"></a>
<FONT color="green">391</FONT>                                               int maxIterations) {<a name="line.391"></a>
<FONT color="green">392</FONT>            double ret;<a name="line.392"></a>
<FONT color="green">393</FONT>    <a name="line.393"></a>
<FONT color="green">394</FONT>            if (Double.isNaN(a) || Double.isNaN(x) || (a &lt;= 0.0) || (x &lt; 0.0)) {<a name="line.394"></a>
<FONT color="green">395</FONT>                ret = Double.NaN;<a name="line.395"></a>
<FONT color="green">396</FONT>            } else if (x == 0.0) {<a name="line.396"></a>
<FONT color="green">397</FONT>                ret = 1.0;<a name="line.397"></a>
<FONT color="green">398</FONT>            } else if (x &lt; a + 1.0) {<a name="line.398"></a>
<FONT color="green">399</FONT>                // use regularizedGammaP because it should converge faster in this<a name="line.399"></a>
<FONT color="green">400</FONT>                // case.<a name="line.400"></a>
<FONT color="green">401</FONT>                ret = 1.0 - regularizedGammaP(a, x, epsilon, maxIterations);<a name="line.401"></a>
<FONT color="green">402</FONT>            } else {<a name="line.402"></a>
<FONT color="green">403</FONT>                // create continued fraction<a name="line.403"></a>
<FONT color="green">404</FONT>                ContinuedFraction cf = new ContinuedFraction() {<a name="line.404"></a>
<FONT color="green">405</FONT>    <a name="line.405"></a>
<FONT color="green">406</FONT>                    @Override<a name="line.406"></a>
<FONT color="green">407</FONT>                    protected double getA(int n, double x) {<a name="line.407"></a>
<FONT color="green">408</FONT>                        return ((2.0 * n) + 1.0) - a + x;<a name="line.408"></a>
<FONT color="green">409</FONT>                    }<a name="line.409"></a>
<FONT color="green">410</FONT>    <a name="line.410"></a>
<FONT color="green">411</FONT>                    @Override<a name="line.411"></a>
<FONT color="green">412</FONT>                    protected double getB(int n, double x) {<a name="line.412"></a>
<FONT color="green">413</FONT>                        return n * (a - n);<a name="line.413"></a>
<FONT color="green">414</FONT>                    }<a name="line.414"></a>
<FONT color="green">415</FONT>                };<a name="line.415"></a>
<FONT color="green">416</FONT>    <a name="line.416"></a>
<FONT color="green">417</FONT>                ret = 1.0 / cf.evaluate(x, epsilon, maxIterations);<a name="line.417"></a>
<FONT color="green">418</FONT>                ret = FastMath.exp(-x + (a * FastMath.log(x)) - logGamma(a)) * ret;<a name="line.418"></a>
<FONT color="green">419</FONT>            }<a name="line.419"></a>
<FONT color="green">420</FONT>    <a name="line.420"></a>
<FONT color="green">421</FONT>            return ret;<a name="line.421"></a>
<FONT color="green">422</FONT>        }<a name="line.422"></a>
<FONT color="green">423</FONT>    <a name="line.423"></a>
<FONT color="green">424</FONT>    <a name="line.424"></a>
<FONT color="green">425</FONT>        /**<a name="line.425"></a>
<FONT color="green">426</FONT>         * &lt;p&gt;Computes the digamma function of x.&lt;/p&gt;<a name="line.426"></a>
<FONT color="green">427</FONT>         *<a name="line.427"></a>
<FONT color="green">428</FONT>         * &lt;p&gt;This is an independently written implementation of the algorithm described in<a name="line.428"></a>
<FONT color="green">429</FONT>         * Jose Bernardo, Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976.&lt;/p&gt;<a name="line.429"></a>
<FONT color="green">430</FONT>         *<a name="line.430"></a>
<FONT color="green">431</FONT>         * &lt;p&gt;Some of the constants have been changed to increase accuracy at the moderate expense<a name="line.431"></a>
<FONT color="green">432</FONT>         * of run-time.  The result should be accurate to within 10^-8 absolute tolerance for<a name="line.432"></a>
<FONT color="green">433</FONT>         * x &gt;= 10^-5 and within 10^-8 relative tolerance for x &gt; 0.&lt;/p&gt;<a name="line.433"></a>
<FONT color="green">434</FONT>         *<a name="line.434"></a>
<FONT color="green">435</FONT>         * &lt;p&gt;Performance for large negative values of x will be quite expensive (proportional to<a name="line.435"></a>
<FONT color="green">436</FONT>         * |x|).  Accuracy for negative values of x should be about 10^-8 absolute for results<a name="line.436"></a>
<FONT color="green">437</FONT>         * less than 10^5 and 10^-8 relative for results larger than that.&lt;/p&gt;<a name="line.437"></a>
<FONT color="green">438</FONT>         *<a name="line.438"></a>
<FONT color="green">439</FONT>         * @param x Argument.<a name="line.439"></a>
<FONT color="green">440</FONT>         * @return digamma(x) to within 10-8 relative or absolute error whichever is smaller.<a name="line.440"></a>
<FONT color="green">441</FONT>         * @see &lt;a href="http://en.wikipedia.org/wiki/Digamma_function"&gt;Digamma&lt;/a&gt;<a name="line.441"></a>
<FONT color="green">442</FONT>         * @see &lt;a href="http://www.uv.es/~bernardo/1976AppStatist.pdf"&gt;Bernardo&amp;apos;s original article &lt;/a&gt;<a name="line.442"></a>
<FONT color="green">443</FONT>         * @since 2.0<a name="line.443"></a>
<FONT color="green">444</FONT>         */<a name="line.444"></a>
<FONT color="green">445</FONT>        public static double digamma(double x) {<a name="line.445"></a>
<FONT color="green">446</FONT>            if (x &gt; 0 &amp;&amp; x &lt;= S_LIMIT) {<a name="line.446"></a>
<FONT color="green">447</FONT>                // use method 5 from Bernardo AS103<a name="line.447"></a>
<FONT color="green">448</FONT>                // accurate to O(x)<a name="line.448"></a>
<FONT color="green">449</FONT>                return -GAMMA - 1 / x;<a name="line.449"></a>
<FONT color="green">450</FONT>            }<a name="line.450"></a>
<FONT color="green">451</FONT>    <a name="line.451"></a>
<FONT color="green">452</FONT>            if (x &gt;= C_LIMIT) {<a name="line.452"></a>
<FONT color="green">453</FONT>                // use method 4 (accurate to O(1/x^8)<a name="line.453"></a>
<FONT color="green">454</FONT>                double inv = 1 / (x * x);<a name="line.454"></a>
<FONT color="green">455</FONT>                //            1       1        1         1<a name="line.455"></a>
<FONT color="green">456</FONT>                // log(x) -  --- - ------ + ------- - -------<a name="line.456"></a>
<FONT color="green">457</FONT>                //           2 x   12 x^2   120 x^4   252 x^6<a name="line.457"></a>
<FONT color="green">458</FONT>                return FastMath.log(x) - 0.5 / x - inv * ((1.0 / 12) + inv * (1.0 / 120 - inv / 252));<a name="line.458"></a>
<FONT color="green">459</FONT>            }<a name="line.459"></a>
<FONT color="green">460</FONT>    <a name="line.460"></a>
<FONT color="green">461</FONT>            return digamma(x + 1) - 1 / x;<a name="line.461"></a>
<FONT color="green">462</FONT>        }<a name="line.462"></a>
<FONT color="green">463</FONT>    <a name="line.463"></a>
<FONT color="green">464</FONT>        /**<a name="line.464"></a>
<FONT color="green">465</FONT>         * Computes the trigamma function of x.<a name="line.465"></a>
<FONT color="green">466</FONT>         * This function is derived by taking the derivative of the implementation<a name="line.466"></a>
<FONT color="green">467</FONT>         * of digamma.<a name="line.467"></a>
<FONT color="green">468</FONT>         *<a name="line.468"></a>
<FONT color="green">469</FONT>         * @param x Argument.<a name="line.469"></a>
<FONT color="green">470</FONT>         * @return trigamma(x) to within 10-8 relative or absolute error whichever is smaller<a name="line.470"></a>
<FONT color="green">471</FONT>         * @see &lt;a href="http://en.wikipedia.org/wiki/Trigamma_function"&gt;Trigamma&lt;/a&gt;<a name="line.471"></a>
<FONT color="green">472</FONT>         * @see Gamma#digamma(double)<a name="line.472"></a>
<FONT color="green">473</FONT>         * @since 2.0<a name="line.473"></a>
<FONT color="green">474</FONT>         */<a name="line.474"></a>
<FONT color="green">475</FONT>        public static double trigamma(double x) {<a name="line.475"></a>
<FONT color="green">476</FONT>            if (x &gt; 0 &amp;&amp; x &lt;= S_LIMIT) {<a name="line.476"></a>
<FONT color="green">477</FONT>                return 1 / (x * x);<a name="line.477"></a>
<FONT color="green">478</FONT>            }<a name="line.478"></a>
<FONT color="green">479</FONT>    <a name="line.479"></a>
<FONT color="green">480</FONT>            if (x &gt;= C_LIMIT) {<a name="line.480"></a>
<FONT color="green">481</FONT>                double inv = 1 / (x * x);<a name="line.481"></a>
<FONT color="green">482</FONT>                //  1    1      1       1       1<a name="line.482"></a>
<FONT color="green">483</FONT>                //  - + ---- + ---- - ----- + -----<a name="line.483"></a>
<FONT color="green">484</FONT>                //  x      2      3       5       7<a name="line.484"></a>
<FONT color="green">485</FONT>                //      2 x    6 x    30 x    42 x<a name="line.485"></a>
<FONT color="green">486</FONT>                return 1 / x + inv / 2 + inv / x * (1.0 / 6 - inv * (1.0 / 30 + inv / 42));<a name="line.486"></a>
<FONT color="green">487</FONT>            }<a name="line.487"></a>
<FONT color="green">488</FONT>    <a name="line.488"></a>
<FONT color="green">489</FONT>            return trigamma(x + 1) + 1 / (x * x);<a name="line.489"></a>
<FONT color="green">490</FONT>        }<a name="line.490"></a>
<FONT color="green">491</FONT>    <a name="line.491"></a>
<FONT color="green">492</FONT>        /**<a name="line.492"></a>
<FONT color="green">493</FONT>         * &lt;p&gt;<a name="line.493"></a>
<FONT color="green">494</FONT>         * Returns the Lanczos approximation used to compute the gamma function.<a name="line.494"></a>
<FONT color="green">495</FONT>         * The Lanczos approximation is related to the Gamma function by the<a name="line.495"></a>
<FONT color="green">496</FONT>         * following equation<a name="line.496"></a>
<FONT color="green">497</FONT>         * &lt;center&gt;<a name="line.497"></a>
<FONT color="green">498</FONT>         * {@code gamma(x) = sqrt(2 * pi) / x * (x + g + 0.5) ^ (x + 0.5)<a name="line.498"></a>
<FONT color="green">499</FONT>         *                   * exp(-x - g - 0.5) * lanczos(x)},<a name="line.499"></a>
<FONT color="green">500</FONT>         * &lt;/center&gt;<a name="line.500"></a>
<FONT color="green">501</FONT>         * where {@code g} is the Lanczos constant.<a name="line.501"></a>
<FONT color="green">502</FONT>         * &lt;/p&gt;<a name="line.502"></a>
<FONT color="green">503</FONT>         *<a name="line.503"></a>
<FONT color="green">504</FONT>         * @param x Argument.<a name="line.504"></a>
<FONT color="green">505</FONT>         * @return The Lanczos approximation.<a name="line.505"></a>
<FONT color="green">506</FONT>         * @see &lt;a href="http://mathworld.wolfram.com/LanczosApproximation.html"&gt;Lanczos Approximation&lt;/a&gt;<a name="line.506"></a>
<FONT color="green">507</FONT>         * equations (1) through (5), and Paul Godfrey's<a name="line.507"></a>
<FONT color="green">508</FONT>         * &lt;a href="http://my.fit.edu/~gabdo/gamma.txt"&gt;Note on the computation<a name="line.508"></a>
<FONT color="green">509</FONT>         * of the convergent Lanczos complex Gamma approximation&lt;/a&gt;<a name="line.509"></a>
<FONT color="green">510</FONT>         * @since 3.1<a name="line.510"></a>
<FONT color="green">511</FONT>         */<a name="line.511"></a>
<FONT color="green">512</FONT>        public static double lanczos(final double x) {<a name="line.512"></a>
<FONT color="green">513</FONT>            double sum = 0.0;<a name="line.513"></a>
<FONT color="green">514</FONT>            for (int i = LANCZOS.length - 1; i &gt; 0; --i) {<a name="line.514"></a>
<FONT color="green">515</FONT>                sum = sum + (LANCZOS[i] / (x + i));<a name="line.515"></a>
<FONT color="green">516</FONT>            }<a name="line.516"></a>
<FONT color="green">517</FONT>            return sum + LANCZOS[0];<a name="line.517"></a>
<FONT color="green">518</FONT>        }<a name="line.518"></a>
<FONT color="green">519</FONT>    <a name="line.519"></a>
<FONT color="green">520</FONT>        /**<a name="line.520"></a>
<FONT color="green">521</FONT>         * Returns the value of 1 / &amp;Gamma;(1 + x) - 1 for -0&amp;#46;5 &amp;le; x &amp;le;<a name="line.521"></a>
<FONT color="green">522</FONT>         * 1&amp;#46;5. This implementation is based on the double precision<a name="line.522"></a>
<FONT color="green">523</FONT>         * implementation in the &lt;em&gt;NSWC Library of Mathematics Subroutines&lt;/em&gt;,<a name="line.523"></a>
<FONT color="green">524</FONT>         * {@code DGAM1}.<a name="line.524"></a>
<FONT color="green">525</FONT>         *<a name="line.525"></a>
<FONT color="green">526</FONT>         * @param x Argument.<a name="line.526"></a>
<FONT color="green">527</FONT>         * @return The value of {@code 1.0 / Gamma(1.0 + x) - 1.0}.<a name="line.527"></a>
<FONT color="green">528</FONT>         * @throws NumberIsTooSmallException if {@code x &lt; -0.5}<a name="line.528"></a>
<FONT color="green">529</FONT>         * @throws NumberIsTooLargeException if {@code x &gt; 1.5}<a name="line.529"></a>
<FONT color="green">530</FONT>         * @since 3.1<a name="line.530"></a>
<FONT color="green">531</FONT>         */<a name="line.531"></a>
<FONT color="green">532</FONT>        public static double invGamma1pm1(final double x) {<a name="line.532"></a>
<FONT color="green">533</FONT>    <a name="line.533"></a>
<FONT color="green">534</FONT>            if (x &lt; -0.5) {<a name="line.534"></a>
<FONT color="green">535</FONT>                throw new NumberIsTooSmallException(x, -0.5, true);<a name="line.535"></a>
<FONT color="green">536</FONT>            }<a name="line.536"></a>
<FONT color="green">537</FONT>            if (x &gt; 1.5) {<a name="line.537"></a>
<FONT color="green">538</FONT>                throw new NumberIsTooLargeException(x, 1.5, true);<a name="line.538"></a>
<FONT color="green">539</FONT>            }<a name="line.539"></a>
<FONT color="green">540</FONT>    <a name="line.540"></a>
<FONT color="green">541</FONT>            final double ret;<a name="line.541"></a>
<FONT color="green">542</FONT>            final double t = x &lt;= 0.5 ? x : (x - 0.5) - 0.5;<a name="line.542"></a>
<FONT color="green">543</FONT>            if (t &lt; 0.0) {<a name="line.543"></a>
<FONT color="green">544</FONT>                final double a = INV_GAMMA1P_M1_A0 + t * INV_GAMMA1P_M1_A1;<a name="line.544"></a>
<FONT color="green">545</FONT>                double b = INV_GAMMA1P_M1_B8;<a name="line.545"></a>
<FONT color="green">546</FONT>                b = INV_GAMMA1P_M1_B7 + t * b;<a name="line.546"></a>
<FONT color="green">547</FONT>                b = INV_GAMMA1P_M1_B6 + t * b;<a name="line.547"></a>
<FONT color="green">548</FONT>                b = INV_GAMMA1P_M1_B5 + t * b;<a name="line.548"></a>
<FONT color="green">549</FONT>                b = INV_GAMMA1P_M1_B4 + t * b;<a name="line.549"></a>
<FONT color="green">550</FONT>                b = INV_GAMMA1P_M1_B3 + t * b;<a name="line.550"></a>
<FONT color="green">551</FONT>                b = INV_GAMMA1P_M1_B2 + t * b;<a name="line.551"></a>
<FONT color="green">552</FONT>                b = INV_GAMMA1P_M1_B1 + t * b;<a name="line.552"></a>
<FONT color="green">553</FONT>                b = 1.0 + t * b;<a name="line.553"></a>
<FONT color="green">554</FONT>    <a name="line.554"></a>
<FONT color="green">555</FONT>                double c = INV_GAMMA1P_M1_C13 + t * (a / b);<a name="line.555"></a>
<FONT color="green">556</FONT>                c = INV_GAMMA1P_M1_C12 + t * c;<a name="line.556"></a>
<FONT color="green">557</FONT>                c = INV_GAMMA1P_M1_C11 + t * c;<a name="line.557"></a>
<FONT color="green">558</FONT>                c = INV_GAMMA1P_M1_C10 + t * c;<a name="line.558"></a>
<FONT color="green">559</FONT>                c = INV_GAMMA1P_M1_C9 + t * c;<a name="line.559"></a>
<FONT color="green">560</FONT>                c = INV_GAMMA1P_M1_C8 + t * c;<a name="line.560"></a>
<FONT color="green">561</FONT>                c = INV_GAMMA1P_M1_C7 + t * c;<a name="line.561"></a>
<FONT color="green">562</FONT>                c = INV_GAMMA1P_M1_C6 + t * c;<a name="line.562"></a>
<FONT color="green">563</FONT>                c = INV_GAMMA1P_M1_C5 + t * c;<a name="line.563"></a>
<FONT color="green">564</FONT>                c = INV_GAMMA1P_M1_C4 + t * c;<a name="line.564"></a>
<FONT color="green">565</FONT>                c = INV_GAMMA1P_M1_C3 + t * c;<a name="line.565"></a>
<FONT color="green">566</FONT>                c = INV_GAMMA1P_M1_C2 + t * c;<a name="line.566"></a>
<FONT color="green">567</FONT>                c = INV_GAMMA1P_M1_C1 + t * c;<a name="line.567"></a>
<FONT color="green">568</FONT>                c = INV_GAMMA1P_M1_C + t * c;<a name="line.568"></a>
<FONT color="green">569</FONT>                if (x &gt; 0.5) {<a name="line.569"></a>
<FONT color="green">570</FONT>                    ret = t * c / x;<a name="line.570"></a>
<FONT color="green">571</FONT>                } else {<a name="line.571"></a>
<FONT color="green">572</FONT>                    ret = x * ((c + 0.5) + 0.5);<a name="line.572"></a>
<FONT color="green">573</FONT>                }<a name="line.573"></a>
<FONT color="green">574</FONT>            } else {<a name="line.574"></a>
<FONT color="green">575</FONT>                double p = INV_GAMMA1P_M1_P6;<a name="line.575"></a>
<FONT color="green">576</FONT>                p = INV_GAMMA1P_M1_P5 + t * p;<a name="line.576"></a>
<FONT color="green">577</FONT>                p = INV_GAMMA1P_M1_P4 + t * p;<a name="line.577"></a>
<FONT color="green">578</FONT>                p = INV_GAMMA1P_M1_P3 + t * p;<a name="line.578"></a>
<FONT color="green">579</FONT>                p = INV_GAMMA1P_M1_P2 + t * p;<a name="line.579"></a>
<FONT color="green">580</FONT>                p = INV_GAMMA1P_M1_P1 + t * p;<a name="line.580"></a>
<FONT color="green">581</FONT>                p = INV_GAMMA1P_M1_P0 + t * p;<a name="line.581"></a>
<FONT color="green">582</FONT>    <a name="line.582"></a>
<FONT color="green">583</FONT>                double q = INV_GAMMA1P_M1_Q4;<a name="line.583"></a>
<FONT color="green">584</FONT>                q = INV_GAMMA1P_M1_Q3 + t * q;<a name="line.584"></a>
<FONT color="green">585</FONT>                q = INV_GAMMA1P_M1_Q2 + t * q;<a name="line.585"></a>
<FONT color="green">586</FONT>                q = INV_GAMMA1P_M1_Q1 + t * q;<a name="line.586"></a>
<FONT color="green">587</FONT>                q = 1.0 + t * q;<a name="line.587"></a>
<FONT color="green">588</FONT>    <a name="line.588"></a>
<FONT color="green">589</FONT>                double c = INV_GAMMA1P_M1_C13 + (p / q) * t;<a name="line.589"></a>
<FONT color="green">590</FONT>                c = INV_GAMMA1P_M1_C12 + t * c;<a name="line.590"></a>
<FONT color="green">591</FONT>                c = INV_GAMMA1P_M1_C11 + t * c;<a name="line.591"></a>
<FONT color="green">592</FONT>                c = INV_GAMMA1P_M1_C10 + t * c;<a name="line.592"></a>
<FONT color="green">593</FONT>                c = INV_GAMMA1P_M1_C9 + t * c;<a name="line.593"></a>
<FONT color="green">594</FONT>                c = INV_GAMMA1P_M1_C8 + t * c;<a name="line.594"></a>
<FONT color="green">595</FONT>                c = INV_GAMMA1P_M1_C7 + t * c;<a name="line.595"></a>
<FONT color="green">596</FONT>                c = INV_GAMMA1P_M1_C6 + t * c;<a name="line.596"></a>
<FONT color="green">597</FONT>                c = INV_GAMMA1P_M1_C5 + t * c;<a name="line.597"></a>
<FONT color="green">598</FONT>                c = INV_GAMMA1P_M1_C4 + t * c;<a name="line.598"></a>
<FONT color="green">599</FONT>                c = INV_GAMMA1P_M1_C3 + t * c;<a name="line.599"></a>
<FONT color="green">600</FONT>                c = INV_GAMMA1P_M1_C2 + t * c;<a name="line.600"></a>
<FONT color="green">601</FONT>                c = INV_GAMMA1P_M1_C1 + t * c;<a name="line.601"></a>
<FONT color="green">602</FONT>                c = INV_GAMMA1P_M1_C0 + t * c;<a name="line.602"></a>
<FONT color="green">603</FONT>    <a name="line.603"></a>
<FONT color="green">604</FONT>                if (x &gt; 0.5) {<a name="line.604"></a>
<FONT color="green">605</FONT>                    ret = (t / x) * ((c - 0.5) - 0.5);<a name="line.605"></a>
<FONT color="green">606</FONT>                } else {<a name="line.606"></a>
<FONT color="green">607</FONT>                    ret = x * c;<a name="line.607"></a>
<FONT color="green">608</FONT>                }<a name="line.608"></a>
<FONT color="green">609</FONT>            }<a name="line.609"></a>
<FONT color="green">610</FONT>    <a name="line.610"></a>
<FONT color="green">611</FONT>            return ret;<a name="line.611"></a>
<FONT color="green">612</FONT>        }<a name="line.612"></a>
<FONT color="green">613</FONT>    <a name="line.613"></a>
<FONT color="green">614</FONT>        /**<a name="line.614"></a>
<FONT color="green">615</FONT>         * Returns the value of log &amp;Gamma;(1 + x) for -0&amp;#46;5 &amp;le; x &amp;le; 1&amp;#46;5.<a name="line.615"></a>
<FONT color="green">616</FONT>         * This implementation is based on the double precision implementation in<a name="line.616"></a>
<FONT color="green">617</FONT>         * the &lt;em&gt;NSWC Library of Mathematics Subroutines&lt;/em&gt;, {@code DGMLN1}.<a name="line.617"></a>
<FONT color="green">618</FONT>         *<a name="line.618"></a>
<FONT color="green">619</FONT>         * @param x Argument.<a name="line.619"></a>
<FONT color="green">620</FONT>         * @return The value of {@code log(Gamma(1 + x))}.<a name="line.620"></a>
<FONT color="green">621</FONT>         * @throws NumberIsTooSmallException if {@code x &lt; -0.5}.<a name="line.621"></a>
<FONT color="green">622</FONT>         * @throws NumberIsTooLargeException if {@code x &gt; 1.5}.<a name="line.622"></a>
<FONT color="green">623</FONT>         * @since 3.1<a name="line.623"></a>
<FONT color="green">624</FONT>         */<a name="line.624"></a>
<FONT color="green">625</FONT>        public static double logGamma1p(final double x)<a name="line.625"></a>
<FONT color="green">626</FONT>            throws NumberIsTooSmallException, NumberIsTooLargeException {<a name="line.626"></a>
<FONT color="green">627</FONT>    <a name="line.627"></a>
<FONT color="green">628</FONT>            if (x &lt; -0.5) {<a name="line.628"></a>
<FONT color="green">629</FONT>                throw new NumberIsTooSmallException(x, -0.5, true);<a name="line.629"></a>
<FONT color="green">630</FONT>            }<a name="line.630"></a>
<FONT color="green">631</FONT>            if (x &gt; 1.5) {<a name="line.631"></a>
<FONT color="green">632</FONT>                throw new NumberIsTooLargeException(x, 1.5, true);<a name="line.632"></a>
<FONT color="green">633</FONT>            }<a name="line.633"></a>
<FONT color="green">634</FONT>    <a name="line.634"></a>
<FONT color="green">635</FONT>            return -FastMath.log1p(invGamma1pm1(x));<a name="line.635"></a>
<FONT color="green">636</FONT>        }<a name="line.636"></a>
<FONT color="green">637</FONT>    <a name="line.637"></a>
<FONT color="green">638</FONT>    <a name="line.638"></a>
<FONT color="green">639</FONT>        /**<a name="line.639"></a>
<FONT color="green">640</FONT>         * Returns the value of Γ(x). Based on the &lt;em&gt;NSWC Library of<a name="line.640"></a>
<FONT color="green">641</FONT>         * Mathematics Subroutines&lt;/em&gt; double precision implementation,<a name="line.641"></a>
<FONT color="green">642</FONT>         * {@code DGAMMA}.<a name="line.642"></a>
<FONT color="green">643</FONT>         *<a name="line.643"></a>
<FONT color="green">644</FONT>         * @param x Argument.<a name="line.644"></a>
<FONT color="green">645</FONT>         * @return the value of {@code Gamma(x)}.<a name="line.645"></a>
<FONT color="green">646</FONT>         * @since 3.1<a name="line.646"></a>
<FONT color="green">647</FONT>         */<a name="line.647"></a>
<FONT color="green">648</FONT>        public static double gamma(final double x) {<a name="line.648"></a>
<FONT color="green">649</FONT>    <a name="line.649"></a>
<FONT color="green">650</FONT>            if ((x == FastMath.rint(x)) &amp;&amp; (x &lt;= 0.0)) {<a name="line.650"></a>
<FONT color="green">651</FONT>                return Double.NaN;<a name="line.651"></a>
<FONT color="green">652</FONT>            }<a name="line.652"></a>
<FONT color="green">653</FONT>    <a name="line.653"></a>
<FONT color="green">654</FONT>            final double ret;<a name="line.654"></a>
<FONT color="green">655</FONT>            final double absX = FastMath.abs(x);<a name="line.655"></a>
<FONT color="green">656</FONT>            if (absX &lt;= 20.0) {<a name="line.656"></a>
<FONT color="green">657</FONT>                if (x &gt;= 1.0) {<a name="line.657"></a>
<FONT color="green">658</FONT>                    /*<a name="line.658"></a>
<FONT color="green">659</FONT>                     * From the recurrence relation<a name="line.659"></a>
<FONT color="green">660</FONT>                     * Gamma(x) = (x - 1) * ... * (x - n) * Gamma(x - n),<a name="line.660"></a>
<FONT color="green">661</FONT>                     * then<a name="line.661"></a>
<FONT color="green">662</FONT>                     * Gamma(t) = 1 / [1 + invGamma1pm1(t - 1)],<a name="line.662"></a>
<FONT color="green">663</FONT>                     * where t = x - n. This means that t must satisfy<a name="line.663"></a>
<FONT color="green">664</FONT>                     * -0.5 &lt;= t - 1 &lt;= 1.5.<a name="line.664"></a>
<FONT color="green">665</FONT>                     */<a name="line.665"></a>
<FONT color="green">666</FONT>                    double prod = 1.0;<a name="line.666"></a>
<FONT color="green">667</FONT>                    double t = x;<a name="line.667"></a>
<FONT color="green">668</FONT>                    while (t &gt; 2.5) {<a name="line.668"></a>
<FONT color="green">669</FONT>                        t = t - 1.0;<a name="line.669"></a>
<FONT color="green">670</FONT>                        prod *= t;<a name="line.670"></a>
<FONT color="green">671</FONT>                    }<a name="line.671"></a>
<FONT color="green">672</FONT>                    ret = prod / (1.0 + invGamma1pm1(t - 1.0));<a name="line.672"></a>
<FONT color="green">673</FONT>                } else {<a name="line.673"></a>
<FONT color="green">674</FONT>                    /*<a name="line.674"></a>
<FONT color="green">675</FONT>                     * From the recurrence relation<a name="line.675"></a>
<FONT color="green">676</FONT>                     * Gamma(x) = Gamma(x + n + 1) / [x * (x + 1) * ... * (x + n)]<a name="line.676"></a>
<FONT color="green">677</FONT>                     * then<a name="line.677"></a>
<FONT color="green">678</FONT>                     * Gamma(x + n + 1) = 1 / [1 + invGamma1pm1(x + n)],<a name="line.678"></a>
<FONT color="green">679</FONT>                     * which requires -0.5 &lt;= x + n &lt;= 1.5.<a name="line.679"></a>
<FONT color="green">680</FONT>                     */<a name="line.680"></a>
<FONT color="green">681</FONT>                    double prod = x;<a name="line.681"></a>
<FONT color="green">682</FONT>                    double t = x;<a name="line.682"></a>
<FONT color="green">683</FONT>                    while (t &lt; -0.5) {<a name="line.683"></a>
<FONT color="green">684</FONT>                        t = t + 1.0;<a name="line.684"></a>
<FONT color="green">685</FONT>                        prod *= t;<a name="line.685"></a>
<FONT color="green">686</FONT>                    }<a name="line.686"></a>
<FONT color="green">687</FONT>                    ret = 1.0 / (prod * (1.0 + invGamma1pm1(t)));<a name="line.687"></a>
<FONT color="green">688</FONT>                }<a name="line.688"></a>
<FONT color="green">689</FONT>            } else {<a name="line.689"></a>
<FONT color="green">690</FONT>                final double y = absX + LANCZOS_G + 0.5;<a name="line.690"></a>
<FONT color="green">691</FONT>                final double gammaAbs = SQRT_TWO_PI / x *<a name="line.691"></a>
<FONT color="green">692</FONT>                                        FastMath.pow(y, absX + 0.5) *<a name="line.692"></a>
<FONT color="green">693</FONT>                                        FastMath.exp(-y) * lanczos(absX);<a name="line.693"></a>
<FONT color="green">694</FONT>                if (x &gt; 0.0) {<a name="line.694"></a>
<FONT color="green">695</FONT>                    ret = gammaAbs;<a name="line.695"></a>
<FONT color="green">696</FONT>                } else {<a name="line.696"></a>
<FONT color="green">697</FONT>                    /*<a name="line.697"></a>
<FONT color="green">698</FONT>                     * From the reflection formula<a name="line.698"></a>
<FONT color="green">699</FONT>                     * Gamma(x) * Gamma(1 - x) * sin(pi * x) = pi,<a name="line.699"></a>
<FONT color="green">700</FONT>                     * and the recurrence relation<a name="line.700"></a>
<FONT color="green">701</FONT>                     * Gamma(1 - x) = -x * Gamma(-x),<a name="line.701"></a>
<FONT color="green">702</FONT>                     * it is found<a name="line.702"></a>
<FONT color="green">703</FONT>                     * Gamma(x) = -pi / [x * sin(pi * x) * Gamma(-x)].<a name="line.703"></a>
<FONT color="green">704</FONT>                     */<a name="line.704"></a>
<FONT color="green">705</FONT>                    ret = -FastMath.PI /<a name="line.705"></a>
<FONT color="green">706</FONT>                          (x * FastMath.sin(FastMath.PI * x) * gammaAbs);<a name="line.706"></a>
<FONT color="green">707</FONT>                }<a name="line.707"></a>
<FONT color="green">708</FONT>            }<a name="line.708"></a>
<FONT color="green">709</FONT>            return ret;<a name="line.709"></a>
<FONT color="green">710</FONT>        }<a name="line.710"></a>
<FONT color="green">711</FONT>    }<a name="line.711"></a>




























































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